Bolyai Gerwein Theorem
1. Show how to cut a trapezoid into two pieces that can be reassembled to form a triangle.
Use this decomposition to verify the formula for the area of a triangle.
2. In all the examples we looked at closed convex polygons. Show that Bolyai-Gerwein is
true for arbritrary polygons (even if they have holes!!!)
3. Show that the Bolyai-Gerwein theorem implies that any two polygons that are equi are
also equicomplementable. Equicomplementable means that you can add congruent
polygons to both figures in such a way that the new figures are congruent.
4. Show that a rectangle and a parallelogram of the same base and height are
equicomplementable.
5. Given an equilateral triangle, show how to cut it into pieces that form a tetrahedron. Now
cut it into pieces that form a triangular prism. How does Bolyai Gerwein inform you
about this problem? How many solutions can you find?
6. Directly show that any two triangles with the same base and area are equi.
7. Show how to cut an arbitrary triangle into pieces that can be rearranged by using
rotations and translations so as to form a mirror image of the triangle. You only need two
cuts.
8. A rectangle is cut out from a bigger rectangle. How can you divide the remaining portion
into two equal parts?
9. What's going on here? (Image taken from: http://www.mathematik.unibielefeld.de/~sillke/)
10. What is the minimum number of cuts you need to turn a triangle, pentagon, hexagon,
heptagon or octagon into a square of the same area? As a fun fact, you can cut a circle
into finitely many pieces to form a square ! (Roughly 1050 pieces are needed).
If you like this kind of thing, checkout
http://mathworld.wolfram.com/Dissection.html
11. Show that every step in the proof of Bolyai Gerwein proof can be done by just using
central symmetries (reflection about a point) and parallel translations.
Hint: The two steps I think presents the most difficulty are 2 and 3. I gave a proof of 2
that presented no problems, but step 3 might be trickier. Come see me or email me at
lalitkumarj@gmail.com if you want a reference.
12. Prove the Pytahgorean theorem by using complementation. Essentially, you need to
show that square c is complementable to the union of square a and b. Hint: use four
copies of the right triangle.
c
a
b
13. Given a rectangular carpet with a hole in it,”mend” it by slicing the carpet into two parts
that can be combined to form a square.
14. Revise the proof of step 4 to show that a polygon with n sides can be broken into n-2
triangles.
15. Use the Hadwiger-Glur Theorem to show that an equilateral triangle and a square of the
same area are not equidecomposable by using parallel translations alone.